Tagged Questions

Lattice is a way of discretizing a quantum field theory for numerical simulations.

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How does lattice field theory relate to string theory?

The former is sort of a "mapping" technique with the goal of making quantum fields computable, while the latter wants to act as an underlying system of it. (as far as I know) What is their relation to ...
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Disclinations, dislocations, lattices, Displacement fields and scaling

I am looking up Frank, and Burger vectors and associated material on dislocation/disclination. It seems straightforward describing a lattice and what dislocation means. It is even possible to restrict ...
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1answer
70 views

Interpretation of $\vec{x}$ in QFT

I am still at an early stage of studying Quantum Field Theory (I am reading QFT In A Nutshell by A. Zee). In the book I'm reading, it starts from a discrete lattice of material "lumps" labeled by ...
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1answer
12 views

Construction of Bound states in Lattice QCD

How the proton state is found and renormalized in lattice QCD? Is there any literature shows the method step by step? What about other bound states in QCD if we somehow know their quantum numbers?
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Unitary gauge in $Z_2$ lattice gauge theory with matter field

A $Z_2$ gauge theory with Ising matter field on a 2-dimensional square lattice has the Hamiltonian \begin{equation} H=-t\sum_{\vec r,j}\sigma_j^x(\vec r)-g\sum_{\vec r}\sigma^z_1(\vec ...
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1answer
51 views

Tight binding operators in 2D lattice system

I have a very naive problem about lattice system, how to translate common operators defined in the bulk ($\hat{x}$, $\hat{p}$...) into their lattice analogues. In a single- band tight-binding ...
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3answers
57 views

What is spin on a lattice site is it electrons or atom as a whole?

Hi I wanted to know what is spin half in lattice site means? Is it electron or atom or total spins half of electrons in a atomic 1d chain or 2d?
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1answer
78 views

What is the advantage of AdS/CFT in studying strong coupled system comparing with the lattice method

I often heard AdS/CFT correspondence provides a powerful framework to study strong coupled system, which perturbation is not applicable. However, lattice method still works in non-perturbative domain. ...
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39 views

Lattice Gas Cellular Automata - HPP model square lattice

In the HPP model of LGCA, a square lattice is used and there is only one collision configuration as mentioned in figure (taken from the book Lattice Gas Cellular Automata and Lattice Boltzmann models ...
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1answer
38 views

Mapping a continuum XY model to a discrete one

A low energy expansion of a system I am currently investigating is described by this XY-like model: $$ H_1 = \int_0^\beta \mathrm{d} \tau \int_{\left[ 0, L \right]^2} \mathrm{d}^2 x \left( J \left( ...
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0answers
31 views

Chern bands and HEP Lattice Fermions: the emergence and the exact map

Chern bands or Chern insulators in 2 spatial dimensional(2D) are a way to construct the bulk insulating gap, but with edge or surfaces with gapless fermions. Such gapless fermions are emergent, and ...
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57 views

Bosonization on the lattice fermion - a rigorous mapping

An inquiry: usually the bosonization is done on the field theory side. The mapping between the fermion operator to the boson operator is done for the field theory operators. As far as we know for the ...
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Degrees of freedom of particles in Lattice Boltzmann method

Is it true that in Lattice Boltzmann method particles have only one degree of freedom even in 3D case? Can someone explain that fact or provide a link? Thanks!
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1answer
111 views

Taylor expansion in Lattice Boltzmann Method derivation

Currently I'm trying to understand Lattice Boltzmann Method for solving CFD problems. In its derivation BGK approximation is used to get rid of complicated collision integral. But when they come to ...
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1answer
26 views

Lattice with snowflakes

Trying to understand these better. I'm trying to find a conventional primitive unit mesh and find the planes of symmetry. Now my attempt was simply finding the next points where it repeated itself. ...
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1answer
92 views

What happens to the free energy of the two-dimensional ising model with vortices?

The classical 2d Ising model has a Hamiltonian of the form: \begin{equation} H = -\sum_{m,n = 0}^{M,N} J_1 x_{m,n}x_{m+1,n} + J_2 x_{m,n}x_{m,n+1} \end{equation} The partition function for the model ...
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29 views

dual variables for lattice fermions

I am quite familiar with duality transformations for lattice spin systems (i.e. systems with global $O(n)$ symmetry) and pure gauge systems (i.e. local $SU(n)$). However, after searching for a bit, I ...
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2answers
381 views

Understanding Elitzur's theorem from Polyakov's simple argument?

I was reading through the first chapter of Polyakov's book "Gauge-fields and Strings" and couldn't understand a hand-wavy argument he makes to explain why in systems with discrete gauge-symmetry only ...
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0answers
37 views

How to get discretization coefficients of matrix $A$ in Finite Volume Method (FVM)? [closed]

First we have Discretization of the Transport Equation $$ \frac{\partial \rho \phi}{\partial t} + \nabla(\rho U \phi) - \nabla (\rho \Gamma_\phi \nabla \phi) = S_\phi (\phi) $$ In Finite Volume ...
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2answers
106 views

Infinitely many points of space-time?

Please can someone provided me with academic literature (Journals/Books, titles & links) which discuss the current view on space time i.e. that there is not Infinitely many points of space-time?
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36 views

Mirrored decoupling fermion doublers and a lattice chiral fermion / gauge theory

Nielsen Ninomiya Fermion-doubling problem has known to be a challenge to construct a chiral fermion or chiral gauge theory on the lattice. There is a proposed resolution to use so-called two mirrored ...
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1answer
281 views

Interpretation of the 1D transverve field Ising model vacuum state in a spin-language

The 1D transverse field Ising model, \begin{equation} H=-J\sum_{i}\sigma_i^z\sigma_{i+1}^z-h\sum_{i}\sigma^x_i, \end{equation} can be solved via the Jordan-Wigner (JW) transformation (for further ...
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Some question on the definition of flux in the projective construction?

Here I have some confusing points about the definition of flux in the projective construction. For example, consider the same mean-field Hamiltonian in my previous question, and assume the $2\times 2$ ...
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1answer
150 views

How many kinds of topological degeneracy are there?

Here I want to summarize the various kinds of topological ground-state degeneracy in condensed matter physics and want to know whether there exists any other kind of topological degeneracy. For ...
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1answer
98 views

Wave vector $\vec{k}$ vs position vector $\vec{x}$

My question is about the $k$-vectors in first Brillouin zone. If I am not misunderstood, the relation k = 2π/(Na) tells that when k goes to zero, we are very very far away from the reference atom and ...
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0answers
36 views

Filling ising model with basis

This questions involves how to fill a lattice with ordered basis: Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For ...
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561 views

Hilbert Space of (quantum) Gauge theory

Since quantum Gauge theory is a quantum mechanical theory, whether someone could explain how to construct and write down the Hilbert Space of quantum Gauge theory with spin-S. (Are there something ...
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0answers
108 views

Lattice theory in mathematics and physics

I have undertaken a project examining lattice model and trying to construct algorithm that could work on all lattice (in physical sense, or crystal structure). I notice there is a branch in ...
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2answers
65 views

What does “sites” mean in the lattice language?

I acknowledge that this question is quite trivial. But in the lattice jargon, what does a $N$-sites lattice mean? it's a lattice $N\times N$ or it's a lattice with $N$ vertices? another option ...
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64 views

Crystal, lattice, periodic graph and graph coloring

I am working across mathematics, physics and engineering. And I am looking for whether there exists already formally established knowledge in the field. Given a periodic graph (actually a physical ...
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1answer
148 views

Does a Lagrangian imply a well-defined quantum Hamiltonianian with a Hilbert space?

The question is about: (1) whether giving a Lagrangian is sufficient enough to (uniquely) well-define a Hamiltonianian quantum theory with a Hilbert space? The answer should be Yes, or No. If ...
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1answer
154 views

How to understand the entanglement in a lattice fermion system?

Topological insulator is a fermion system with only short-ranged entanglement, what does the entanglement mean here? For example, the Hilbert space $V_s$ of a lattice $N$ spin-1/2 system is ...
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1answer
212 views

A naive question on the $U(1)$ gauge transformation of electromagnetic field?

For simplicity, in the following we set the electric charge $e=1$ and consider a lattice spinless free electron system in an external static magnetic field $\mathbf{B}=\nabla\times\mathbf{A}$ ...
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95 views

Taylor expansion in classical 1D harmonic chain (classical field theory)

I'm struggling with something apparently really easy but not entirely straightforward. In "Condensed Matter Field Theory" of Altland and Simons the classical 1D harmonic chain is treated as ...
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1answer
162 views

Entanglement entropy for U(1) lattice gauge theory

Can someone please let me know if there is some reference for the calculation of entanglement entropy of U(1) lattice gauge theory? I have seen a few references where Z2 lattice gauge theory has been ...
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1answer
260 views

Anomalies for not-on-site discrete gauge symmetries

If a symmetry group $G$ (let's say finite for simplicity) acts on a lattice theory by acting only on the vertex variables, I will call it ultralocal. Any ultralocal symmetry can be gauged. However, in ...
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1answer
103 views

Probablity density function $f(\mathbf{x},\mathbf{v},t)$

The phase density function is usually denoted as $f(\mathbf{x},\mathbf{v},t)$ which gives probable number of particles moving with velocity $ \mathbf{v}$ at position $\mathbf{x}$ at time t. Also we ...
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1answer
367 views

The Bhatnagar-Gross-Krook (BGK) approximation of the collision integral

Bhatnagar, Gross and Krook (BGK) proposed a relaxation term for the collision integral $ Q$ as follows $$J = \frac{1}{\tau} (f^{eq} - f)$$ where $f^{eq}$ is the distribution at equilibrium. $Q$ has ...
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0answers
80 views

Non-Hermiticity when Fourier transforming onto a finite lattice

I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where $$ H = \sum \limits_{<ij>}a^\dagger_i b_j + h.c $$ Where $i$ belongs to sublattice $A$, and $j$ to ...
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131 views

Lattice model completely constrained by boundary data

I am dealing with a lattice model that has the peculiar property that if I specify all the spins on the boundary, by local conservation laws, the whole lattice configuration (throughout the whole ...
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1answer
106 views

$SU(N)$ Neel manifold

I have seen multiple papers talking about the manifold, $M$ of the Neel order for an $SU(N)$ magnet is $$M~=~\frac{U(N)}{U(m)\times U(N-m)}.$$ So for instance, a $SU(2)$ magnet has manifold $$M ~=~ ...
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1answer
326 views

Expansion in solid spherical harmonics on the lattice

I'm interested in calculating scattering processes (e.g. Coulomb scattering of an electron beam by a single ion) in the context of lattice quantum field theory, and wonder if there is something like ...
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2answers
111 views

What symmetries does a lattice calculation need to preserve?

I've heard that it is impossible to have a properly Lorentz-invariant lattice QFT simulation, as the Lorentz invariance is spoiled by the nonzero lattice distance $a$. I've also heard that there are ...
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2answers
168 views

How local is the stress tensor?

I am confused by the definition of the stress tensor in a crystal (let's say a semi-conductor), I don't see how it could be "more local" than over an unit cell. I know that in field theory the stress ...
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0answers
57 views

Orientation in GaAs

I can't find the precise definition of what is the orientation of a GaAs lattice. Being the superposition of two fcc lattices (one of Ga, the other of As), I would think that it is the direction of ...
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1answer
239 views

Chiral coupling in string-nets

In Xiao-Gang Wen's review of topological order http://arxiv.org/abs/1210.1281 , he states in footnote 52 that string-nets are so far unable to produce the chiral coupling between the SU(2) gauge boson ...
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1answer
193 views

Origin of Laue equations?

The Bragg condition (by Bragg in 1913) can be derived by the Laue equations that is making use of the Miller indices and all the latice/crystal stuff (so basically it's bringing Bragg's law to more ...
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1answer
941 views

What is the Hubbard-Holstein model?

Please explain as simply as possible what the Hubbard-Holtstein model is and what it is used for.
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0answers
122 views

phi^4 theory on lattice

I was reading the lecture notes from http://nic.desy.de/sites2009/site_nic/content/e44192/e62778/e91179/e91180/hmc_tutorial_eng.pdf I am a little bit confused how points on the lattice gets assigned ...
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2answers
844 views

What is a bulk phase transition?

I have been able to google "bulk phase transition" and get plenty of results that verify that something called a bulk phase transition exists, however, I cannot seem to find a precise definition of ...